Math 110 Review for Exam 2
1. Use the graph of the sixth degree polynomial p(x) below to answer the following.
a. List each zero of f in point form, and state its likely multiplicity (keep in mind this is a 6th degree polynomial).
b. State the y-intercept in point form.
c. Write a possible formula for p(x). You can leave this in factored form. Remember to use your y-intercept to find a,
the leading coefficient.
2. Given the polynomials below, answer the following:
P (x) = 2x3 + 3x2 − 3x − 2
P (x) = x4 + 2x3 − 7x2 − 8x + 12
a. State all possible rational zeros.
b. For each, use synthetic division to show x = 1 is a zero of the function.
c. Write the function as P (x) = (x − 1)Q(x) and finish factoring the function into linear factors.
d. Graph each function.
e. State the intervals on which P (x) ≤ 0
3. Factor the following polynomials into linear factors (real and complex) and find all zeros (real and complex).
P (x) = x4 + 8x2 − 9
p(x) = x4 − 64
4. Find a formula for a third degree polynomial that has zeros 2 and 3i, and has a y-intercept of -9.
Notice problems #5-7 are from HW 7.
5. Consider the function g(x) = −2x−2 + 3, and answer the following.
a. Refer to the function f (x) = 2x , and state what transformations of f is the function g(x) = −2x−2 + 3.
b. Graph f and g below.
c. State the Domain, Range, and Asymptote of g.
d. Find the average rate of change of g on the interval [1, 3].
6. The function below represents a population of fish in a pond t years after a group of fish were initially placed in the
pond (the pond did not have any fish before this group was introduced). Answer the following.
1200
1 + 11e−0.2t
a. How many fish were in group initially placed in the pond, t = 0?
P (t) =
b. What value does the population approach as t → ∞?
7. Given P (x) = x4 − x3 − 11x2 + 9x + 18, answer the following.
a. List all possible rational zeros of P .
b. Using synthetic division, show x = −1 is a zero of P .
c. Using your work from part b, fully factor P into real linear factors.
d. Sketch a graph of P below.
e. Find the intervals for which x4 − x3 − 11x2 + 9x + 18 ≤ 0
8. For the following rational functions, answer the following.
2x2 − 8
2
R(x)
=
x2 + 2x − 3
x2 − 4x
a. Find the zeros of the function.
R(x) =
R(x) =
x2 + 3x
x−2
R(x) =
x2 − x − 6
x2 − 9
b. Find the y-intercept.
c. Find all asymptotes (Vertical, Horizontal, oblique (slant), if any).
d. Sketch a graph of the function.
e. State the intervals in which R(x) ≥ 0.
9. Given the following graphs of rational functions, answer the following.
a. Find the zeros of the function.
b. Find the y-intercept.
c. Find all asymptotes (Vertical, Horizontal, oblique (slant), if any).
d. What is the domain and range of R(x)?
e. Use your answers above to find a possible formula for R(x).
10. Graph the following exponential functions state the domain, range, and asymptote.
!x
1
g(x) = 4 +
F (x) = 2x − 3
f (x) = 2x−4 + 1
2
h(x) = 6 − 3x
11. Graph f (x) = ex , and then use this to sketch a graph of the following.
g(x) = e−x − 3
h(x) = 1 − ex+1
12. A sky diver jumps from a reasonable height above ground. The downward velocity of the sky diver at time t is given
ft
.
by v(t) = 180(1 − e−0.2t ), where t is measured in seconds and v(t) is measured in feet per second, sec
a. Find the initial velocity of the sky diver.
b. Find the velocity after 10 seconds. Leave your answer exact.
c. The maximum velocity of a falling object with wind resistance is called its terminal velocity. Find the terminal
velocity of the sky diver. To do so, find what the velocity approaches as t → ∞.
13. If $10,000 is invested at an interest rate of 5% per year, find the amount of the investment after 10 years for the
following compounding methods. Leave your answer exact.
a. Annually
b. Monthly
c. Continuously